A 501(c)(3) organization is one that is organized exclusively for religious, charitable, scientific, or educational purposes. Such organizations may advocate or provide educational material consistent with their purposes as long as it does not participate in, or intervene in any political campaign.
A 501(c)(6) is a "business league" of individuals similar to a trade association of organizations. There could be a 501(c)(6) of public health workers, but it would by necessity, have to be limited to issues related exclusively with the welfare of public health workers and not public health issues broadly or generally.